Simple implementation of high fidelity controlled-$i$swap gates and quantum circuit exponentiation of non-Hermitian gates

The $i$swap gate is an entangling swapping gate where the qubits obtain a phase of $i$ if the state of the qubits is swapped. Here we present a simple implementation of the controlled-$i$swap gate. The gate can be implemented with several controls and works by applying a single flux pulse. The gate time is independent of the number of controls, and we find high fidelities for any number of controls. We discuss an implementation of the gates using superconducting circuits and present a realistic implementation proposal, where we have taken decoherence noise and fabrication errors on the superconducting chip in to account, by Monte Carlo simulating possible errors. The general idea presented in this paper is, however, not limited to such implementations. Exponentiation of quantum gates is desired in some quantum information schemes and we therefore also present a quantum circuit for probabilistic exponentiating the $i$swap gate and other non-Hermitian gates.

Figure 1

Simulation of the controlled-$i$swap gate for different values of the coupling $J^x$. The blue lines indicates the average fidelity (left $y$ axis), while the straight red line indicates the gate time $T$ (right $y$ axis). The dashed blue line is the average fidelity with a decoherence time of $T_1=T_2=30\mu \mathrm{s}$, while the solid is without decoherence.

Figure 2

Average fidelity as a function of the number of qubits in the ${\mathrm{C}}^{n}i\mathtt{swap}$ gate. The blue square markers indicate the simulation without decoherence, while the round red markers indicates the simulation done with a decoherence time of $T_1=T_2=30\mu \mathrm{s}$. All simulations are done with $J^z/J=5$, i.e., peak fidelity cf. Fig. 1.

Figure 3

Implementation of the controlled-$i$swap gate, i.e., iFredkin gate. (a) shows a schematic representation of the model implementing the controlled-$i$swap gate, with the green spheres (subscript $T1$ and $T2$) representing the target qubits and the blue sphere (subscript 1) representing the control qubit. The red line indicates an interaction which can be turn on and off, how to do this depends on the qubit implementation, see Sec. 3 for an example of how to do this. (b) shows the superconducting circuit yielding the model in (a). The different parts of the system are colored according to their role, as per (a).

Figure 4

Typical derivatives of the gate parameters $J^x$ (a) and $\tilde{\omega }$ (b). The dashed lines indicates the error on the parameters, found using Monte Carlo simulations. In particular, the parameters comes from column 2 in Table 3.

Figure 5

(a) Average fidelity of the Monte Carlo simulation of the gate, as a function of time. All shown simulation are without noise (b) Distribution of fidelities of the simulations at the gate time. The noise is the same as in Sec. 2a.

Figure 6

(a) Schematic representation of the model leading to controlled swapping between three qubits. The green spheres (subscripts 1, 2, and 3) represent the swapping qubits, while the blue spheres (subscripts $C1$, $C2$, and $C3$) represent the ancilla qubits, which controls the swapping. (b) Quantum circuit representations of the model in (a) respectively for times $T=(2m+1)\pi /\left(2J^x\right)$, $m\in \mathbb{Z}$. The top three ancilla qubits controls the swapping and corresponds to the blue spheres, while the lower three qubits corresponds to the green spheres. The filled circles indicates that the ancilla qubits must be the state $|1\rangle$ for the swap to be activated, while the nonfilled circles indicates that the ancilla qubits must be in the state $|1\rangle$; This corresponds to the time evolution operators in Eq. (17).

Figure 7

(a) Schematic representation of the model leading to controlled swapping between four qubits. The green spheres (subscript 1, 2, 3, and 4) represent the swapping qubits, while the blue spheres (subscript $C1$, $C2$, $C3$, and $C4$) represent the ancilla qubits, which controls the swapping. (b) Quantum circuit representations of the model in (a) for times $T=(2m+1)\pi /\left(2J^x\right)$, $m\in \mathbb{Z}$. The top four ancilla qubits controls the swapping and corresponds to the blue spheres, while the lower four qubits corresponds to the green spheres.

Figure 8

Quantum circuit used to exponentiate a matrix $\widehat{T}$ for which ${\widehat{T}}^n=\mathbb{1}$. On top, we have $n-1$ ancilla qubits which are prepared in the state $|\tilde{\phi }\rangle$. Each acts as a conditional for a $\widehat{T}$ operation, and finally they are all measured in the $\left\{\right|\pm \rangle \}$-basis. Note that the $\widehat{T}$ operation does not have to be a two qubit operation, it can operate on $m$ qubits.

Figure 9

Probability of measuring each of the states in Eq. (24).

Figure 10

Implementation of the ${\mathtt{c}}^{2}i\mathtt{swap}$ gate. (a) shows a schematic representation of the model implementing the ${\mathtt{c}}^{2}i\mathtt{swap}$ gate, with the green spheres (subscript $T1$ and $T2$) representing the target qubit and the blue spheres (subscript 1 and 2) representing the control qubits. (b) shows the superconducting circuit yielding the model in (a). The different parts of the system are colored according to their role, as per (a). Note that in this implementation each target qubit are connected to a control qubit, contrary to the proposed implementation where both control qubits are connected to the first target qubit. It is, however, irrelevant which qubit the control qubits are connected to, the controlled $i$swap gate can be created either way.